Cornell College
STA 200 Fall 2025 Block 1
Scatterplots are useful for visualizing the relationship between two numerical variables.
Do life expectancy and total fertility appear to be associated or independent?
They appear to be linearly and negatively associated: as fertility increases, life expectancy decreases.
Was the relationship the same throughout the years, or did it change?
The relationship changed over the years.
Useful for visualizing one numerical variable. Darker colors represent areas where there are more observations.
How would you describe the distribution of GPAs in this data set?
Make sure to say something about the center, shape, and spread of the distribution.
The sample mean, denoted as \(\bar{x}\), can be calculated as:
\[ \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} \]
where \(x_1, x_2, \cdots, x_n\) represent the n observed values.
The population mean is also computed the same way but is denoted as \(\mu\). It is often not possible to calculate \(\mu\) since population data are rarely available.
The sample mean is a sample statistic, and serves as a point estimate of the population mean. This estimate may not be perfect, but if the sample is good (representative of the population), it is usually a pretty good estimate.
Higher bars represent areas where there are more observations, making it a little easier to judge the center and the shape of the distribution.
Stacked dot plot of GPA
Histogram of extracurricular hours
Which one(s) of these histograms are useful? Which reveal too much about the data? Which hide too much?
Does the histogram have a single prominent peak (unimodal), several prominent peaks (bimodal/multimodal), or no apparent peaks (uniform)?
Modality examples
In order to determine modality, step back and imagine a smooth curve over the histogram — imagine that the bars are wooden blocks and you drop a limp spaghetti over them. The shape the spaghetti would take could be viewed as a smooth curve.
Is the histogram right skewed, left skewed, or symmetric?
Skewness examples
Histograms are said to be skewed to the side of the long tail.
Are there any unusual observations or potential outliers?
Outlier examples
How would you describe the shape of the distribution of hours per week students spend on extracurricular activities?
Histogram of extracurricular hours
Unimodal and right skewed, with a potentially unusual observation at 60 hours/week.
unimodal
bimodal
multimodal
uniform
right skew
left skew
symmetric
Which of these variables do you expect to be uniformly distributed?
Answer: birthdays of classmates
Sketch the expected distributions of the following variables:
Come up with a concise way (1-2 sentences) to teach someone how to determine the expected distribution of any variable.
Typical distribution
http://www.youtube.com/watch?v=4B2xOvKFFz4
How useful are centers alone for conveying the true characteristics of a distribution?
Variance is roughly the average squared deviation from the mean.
\[ s^2 = \frac{\sum_{i = 1}^n (x_i - \bar{x})^2}{n - 1} \]
\[ s^2 = \frac{(5 - 6.71)^2 + (9 - 6.71)^2 + \cdots + (7 - 6.71)^2}{217 - 1} = 4.11~hours^2 \]
Why do we use the squared deviation in the calculation of variance?
The standard deviation is the square root of the variance, and has the same units as the data.
\[ s = \sqrt{s^2} \]
The standard deviation of amount of sleep students get per night can be calculated as:
\[
s = \sqrt{4.11} = 2.03~\text{hours}
\]
We can see that all of the data are within 3 standard deviations of the mean.
The median is the value that splits the data in half when ordered in ascending order.
\[0, 1, \textbf{2}, 3, 4\]
If there are an even number of observations, then the median is the average of the two values in the middle.
\[0, 1, \underline{2, 3}, 4, 5 \rightarrow \frac{2 + 3}{2} = \textbf{2.5}\]
Since the median is the midpoint of the data, 50% of the values are below it. Hence, it is also the \(50^{th}\) percentile.
The \(25^{th}\) percentile is also called the first quartile, Q1.
The \(50^{th}\) percentile is also called the median.
The \(75^{th}\) percentile is also called the third quartile, Q3.
Between Q1 and Q3 is the middle 50% of the data. The range these data span is called the interquartile range, or the IQR.
\[ IQR = Q3 - Q1 \]
The box in a box plot represents the middle 50% of the data, and the thick line in the box is the median.
Box plot of study hours
Box plot anatomy
\[ \begin{aligned} \text{max upper whisker reach} &= Q3 + 1.5 \times IQR \\ \text{max lower whisker reach} &= Q1 - 1.5 \times IQR \end{aligned} \]
\[ \begin{aligned} \text{IQR} &= 20 - 10 = 10 \\ \text{max upper whisker reach} &= 20 + 1.5 \times 10 = 35 \\ \text{max lower whisker reach} &= 10 - 1.5 \times 10 = -5 \end{aligned} \]
Question: Why is it important to look for outliers?
Identify extreme skew in the distribution.
Identify data collection and entry errors.
Provide insight into interesting features of the data.
Question: How would sample statistics such as mean, median, SD, and IQR of household income be affected if the largest value was replaced with $10 million? What if the smallest value was replaced with $10 million?
Household income dot plot
Household income dot plot
| scenario | robust | not robust | ||||
|---|---|---|---|---|---|---|
| median | IQR | \(\bar{x}\) | \(s\) | |||
| original data | 190K | 200K | 245K | 226K | ||
| move largest to $10 million | 190K | 200K | 309K | 853K | ||
| move smallest to $10 million | 200K | 200K | 316K | 854K |
Median and IQR are more robust to skewness and outliers than mean and SD. Therefore,
Question: If you would like to estimate the typical household income for a student, would you be more interested in the mean or median income?
Answer: Median
If the distribution is symmetric, center is often defined as the mean: mean ≈ median
If the distribution is skewed or has extreme outliers, center is often defined as the median
Which is most likely true for the distribution of percentage of time actually spent taking notes in class versus on Facebook, Twitter, etc.?
Median: 80%
Mean: 76%
Answer: mean < median
When data are extremely skewed, transforming them might make modeling easier. A common transformation is the log transformation.
The histograms on the left show the distribution of number of basketball games attended by students. The histogram on the right shows the distribution of log of number of games attended.
Skewed data are easier to model with when they are transformed because outliers tend to become far less prominent after an appropriate transformation.
| # of games | 70 | 50 | 25 | ⋯ |
|---|---|---|---|---|
| log(# of games) | 4.25 | 3.91 | 3.22 | ⋯ |
However, results of an analysis in log units of the measured variable might be difficult to interpret.
Q: What other variables would you expect to be extremely skewed?
A: Salary, housing prices, etc.
Q: What patterns are apparent in the change in population between 2000 and 2010?
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